Estimates of the choice numbers and the Ohba numbers of the complete multipartite graphs \(K(m, n, 1, \ldots, 1)\) and \(K(m, n, 2, \ldots, 2)\) are provided for various values of \(m \geq n \geq 1\). The Ohba number of a graph \(G\) is the smallest integer \(n\) such that \(\operatorname{ch}(G \vee K_n) = \chi(G \vee K_n)\).